in the figure Amazon midpoint of QR angle r p r q is equals to 90 degree then prove that PQ square is equals to 4 cm square minus 3 PR
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Step-by-step explanation:
Given :- In ∆PRQ, angle R = 90°
M is midpoint of RQ.
RM = MQ = 1/2 RQ
⚫ To Prove :- PQ^2 = 4PM^2 - 3PR^2
⚫ Proof :- In ∆PRM, angle PRM = 90°
By pythagoras theorem...,
PM^2 = PR^2 + RM^2
PM^2 = PR^2 + ( 1/2 RQ )^2 --( M is midpoint )
PM^2 = PR^2 + RQ^2/4
PM^2 = ( 4PR^2 + RQ^2 )/4
4PM^2 = 4PR^2 + RQ^2
RQ^2 = 4PM^2 - 4PR^2 --- ( 1 )
In ∆PRQ, angle PRQ = 90°
By pythagoras theorem..,
PQ^2 = PR^2 + RQ^2
PQ^2 = PR^2 + ( 4PM^2 - 4PR^2 ) --( From eq.1 )
PQ^2 = PR^2 + 4PM^2 - 4PR^2
PQ^2 = 4PM^2 - 3PR^2
Hence proved PQ^2 = 4PM^2 - 3PR^2
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